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Theorem isfin2 8706
Description: Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin2  |-  ( A  e.  V  ->  ( A  e. FinII 
<-> 
A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) ) )
Distinct variable group:    y, A
Allowed substitution hint:    V( y)

Proof of Theorem isfin2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pweq 3958 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
21pweqd 3960 . . 3  |-  ( x  =  A  ->  ~P ~P x  =  ~P ~P A )
32raleqdv 3010 . 2  |-  ( x  =  A  ->  ( A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y )  <->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) ) )
4 df-fin2 8698 . 2  |- FinII  =  {
x  |  A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) }
53, 4elab2g 3198 1  |-  ( A  e.  V  ->  ( A  e. FinII 
<-> 
A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   (/)c0 3738   ~Pcpw 3955   U.cuni 4191    Or wor 4743   [ C.] crpss 6561  FinIIcfin2 8691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-v 3061  df-in 3421  df-ss 3428  df-pw 3957  df-fin2 8698
This theorem is referenced by:  fin2i  8707  isfin2-2  8731  ssfin2  8732  enfin2i  8733  fin12  8825  fin1a2s  8826
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